Final answer:
The formula for the area of an ellipse is A = πab, where a and b are the lengths of the semi-major and semi-minor axes respectively. This formula can be proven using calculus or by approximating the elliptical shape with a polygon and calculating the area of the individual triangles formed.
Step-by-step explanation:
An ellipse is a geometric shape defined as the set of all points in a plane, the sum of whose distances to two fixed points (called the foci) is constant. The formula for the area of an ellipse is A = πab, where a and b are the lengths of the semi-major and semi-minor axes respectively.
To prove this formula, we can use calculus. We start by considering the ellipse in its standard form, which is x²/a² + y²/b² = 1. We can use elliptical coordinates to parameterize the equation and then integrate it over a suitable range to obtain the area.
Alternatively, we can use the fact that an ellipse can be approximated by a polygon with a large number of sides. By drawing such a polygon inside the ellipse, we can calculate the area of each individual triangle formed by connecting the polygon vertices to the center of the ellipse. Summing up the areas of all the triangles will give us an approximate value for the area of the ellipse.